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OrderReduction - Reduction of Order and Double Roots We...

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Reduction of Order and Double Roots We have found the general solution to the equation ay 00 + by 0 + cy = 0 in the case where the characteristic equation has distinct real roots, or complex roots. It remains to consider the case when we have a double root, that is when the real roots are equal (when b 2 - 4 ac = 0 ). In this case we have run into a common problem, which is that we know one solution is y 1 ( x ) = e rx where r is the double root of the characteristic equation, but our last theorem states that we need another linearly independent solution y 2 ( x ) before we can write down the general solution. Reduction of Order The problem, generally stated is this: Knowing one solution y 1 ( x ) to a second-order linear differential equation, is there a systematic way to find a linearly independent solution y 2 ( x ) ? This problem is resolved using a method called reduction of order. Suppose that y 1 ( x ) is a solution to the equation y 00 ( x ) + p ( x ) y 0 + q ( x ) y = 0 Let v = v ( x ) be a function of x . We will investigate under what conditions the product y ( x ) = v ( x ) y
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